Abstract: Tof ∈ℂ[x 1…,x n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f −1{0} inf −1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x 1,x 2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.

Abstract: From wave equation to trace formula singularities of the theta function determinant representation of the Selberg zeta function the functional equations of Selberg and Ruelle zeta functions Dirac operators and form Laplacians.

Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: © Annales de l’institut Fourier, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: JUNESANG CHOI, H.M. SRIVASTAVA AND J.R. QUINELots of formulas for series of zeta function have been developed in many ways.We show how we can apply the theory of the double gamma function, which hasrecently been revived according to the study of determinants of Laplacians, toevaluate some series involving the Riemann zeta function.1. INTRODUCTION

Abstract: A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. These algorithms out perform the standard methods based on Euler-Maclaurin summation, are easier to implement and are easier to analyse.

Abstract: Article history: Received 2 March 2010 Revised 28 April 2010 Communicated by David Goss MSC: 11M06

Abstract: To a polynomial f over a p-adic field K and a character X of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s, f, X), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s, f, X) to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of X does not divide the order of any eigenvalue of monodromy of f, then Z(s, f, X) is holomorphic on C. We prove mainly that if the holomorphy conjecture is true for f(xI . xn-) , then it is true for f(xl . .. x ) + xk with k > 3, and we give some applications.

Abstract: We show that in contrast to the various known analogies, the orders of vanishing of the characteristic p zeta functions introduced by Goss sometimes follow interesting pattems involving base q-digits, providing a challenge to understand them in a general framework as in the classical case. We answer some questions raised by Goss about the connection of the zeta values and periods. We also generalize and simplify the proofs of some results.

Abstract: Explicit formulae for the zeta functions of the zeros of Hahn-Exton and Jackson's q-Bessel functions are derived. They can be regarded as spectral sum rules for some discrete quantum billiards.

Abstract: In this paper we consider L-functions satisfying certain standard conditions, their approximation by Dirichlet polynomials and, especially, lower bounds for the lengths of the polynomials that provide good approximations. 1. Introduction For ~(s) the Riemann zeta-function one has the Dirichlet series representation valid for where s = u + it. By the absolute convergence of this series one sees that, even for x not very large, the Dirichlet polynomial L n-s gives a rather good n2 approximation to ~(s), with a remainder which is o( 1 ) as x -~ oo. This is a nice property, since one would expect the finite sum to be easier to work with for purposes of estimation. However, one is of course more interested in estimating ~(s) in the critical strip 0 u 1. Here the above polynomial still provides [T, § 4.11 ] (at least away from the pole) a useful approximation to ~(s), moreover the smoothed polynomials * Supported in part by NSERC Grant A5123. Pervenuto alla Redazione il 2 Luglio 1994. 518 do an even better job, but all of these only for x > (1 + 0 ( 1 )) l!l, and this limits their usefulness for application. 27r Thus the question arises whether shorter approximations of the same quality to ~(s), or for that matter to L-series and general zeta functions, exist. In this paper we investigate approximations by Dirichlet polynomials to L-functions of a fairly general type, and show in many cases that it is not possible to achieve a very good level of approximation by means of polynomials essentially shorter than the known approximations. Thus we may view such a result as a first step toward understanding the analytic complexity of a zeta function. We shall consider L-functions L(s) having the following properties (compare, for example, [S]): (HI) L(s) is given by an absolutely convergent Dirichlet series in the half-plane u > 1, with coefficients an satisfying a, = 1 and an « n°(1). (H2) L(s) is meromorphic of finite order in the whole complex plane, has only finitely many poles and satisfies a functional equation 1 where with constants satisfying From the fact that L(s) is of finite order with finitely many poles and satisfies a functional equation of the above type and from the Phragmen-Lindelof principle, it follows that L(s) has, away from the poles, polynomial growth in any fixed vertical strip. Moreover L(s), for a 1 has order not less than 2 1 and for a 0 has order precisely where It now follows by a well-known argument (cf. [T, § 9.4]) that the number N(T; L) of non-trivial zeros (that is, those not located at the poles of the r factors) of L(s) satisfying 0 t T, is given asymptotically by 1 For a function f (s) we define 7(s)=f(-~). 519 where cL is a constant depending on L. Since we assume a 1 = 1 we may compute the constants explicitly and write this in the form where One should remark that the choice of the parameter Q and the Gamma factors in the above decomposition of are not uniquely determined due to the multiplication formula for the Gamma function. However the key quantities A and CL used in this paper are uniquely determined by L(s). The next assumption that we make about our L-function is that it satisfy a weak zero-density estimate. Let N(u, T; L) denote the number of non-trivial zeros p = {3 + iq of L with 0 q T and {3 > ~ . Then we assume: (H3) For any fixed 6 > 0, we have Our two main results place a limitation on the length of the Dirichlet polynomial (actually, may be replaced by any fixed positive constant) if it is to be a 2 useful approximation to L(s). Specifically, we prove THEOREM 1. Let L(s) satisfy assumptions (Hl)-(H3), and let e,,-’ > 0. Suppose that we have on the segment Our basic strategy is to use a well known lemma of Littlewood to compare, in a suitable rectangle, the number of zeros of the function L(s) with that of the approximating polynomial These should be nearly equal if the approximation is sufficiently good. On the other hand we shall be able to estimate the former using (1.1). This will give a contradiction provided that we can give a smaller upper bound to the number of zeros of D2(s) in case x is not too large. Such a result is provided by the following: 520 PROPOSITION 1. Let Dx(s) given by (1.2) satisfy Then, uniformly for a a oo we have Let also N(a, T, T + H; Dx) denote the number of zeros of D2(s) satisfying u > a, T t T + H, where H T. Then, uniformly for -H a l, we have where the implied constant is absolute. The exponent 2A given in Theorem 1 is sharp, as will be seen in the next section. Nevertheless, a slightly different argument using Rouche’s theorem shows that the bound can be made still more precise if one is willing to strengthen the assumptions to some extent. Specifically, we have: THEOREM 2. Let L(s) satisfy assumptions (HI), (H2), and also, for every 6 > 0, the strengthened zero density bound Suppose that we have on the segment (u = -c’, T t (I + Then x > (1 + o(I»CLT2A, with CL as in ( 1.1’). In Theorem 2, not only the exponent, but even the constant CL is the best possible. We remark that the assumption of (1.3) on the segment with u = -g’ is stronger than the assumption on a corresponding segment with a = 1 -s’; see Proposition 3. In the event that one assumes a stronger version as in Theorem 2, but is willing to settle for the weaker conclusion of Theorem 1, then it is possible to give a somewhat simpler proof which combines the principle of the argument with the result of Proposition 1. Throughout the paper, implied constants may depend on L(s) which is considered to be fixed. It would be of interest to have analogous results that are uniform in the parameter Q. The paper is organized as follows. In Section 2 we give a number of examples illustrating the sharpness of our results. In Section 3 we give an alternative argument that is considerably shorter than the proof of the main 521 theorems, but which gives only weaker bounds except in the case A 1. The remaining sections are devoted to the proof of the Theorems. In Section 4, we give the proof of Proposition 1, bounding the number of zeros of Dirichlet polynomials. In Section 5 we prove Proposition 3 which shows that, given an approximation of the type hypothesized by the theorems, that approximation continues to hold for all larger values of u. In Section 6 we prove a number of consequences of our hypothesis of a zero-density bound. We find, with good localization, thin horizontal strips on which there holds a Lindelof strength bound for L, and within each of these, a horizontal line on which holds a similar bound for L-1. These bounds, which are needed for our application of the Littlewood lemma and the Rouche theorem, improve earlier results which would not have sufficed. Finally, in Section 7, we combine the above preparations to complete the proofs of our results. 2. Some Examples and Remarks EXAMPLE 1. Zeros of Dirichlet polynomials The finite Euler product has length by the prime number theorem and has, for T t 2T, zeros on the imaginary axis at t = 2n7r/ log p. These number again by the prime number theorem. Thus the bound given in Proposition 1 is asymptotically sharp. EXAMPLE 2. Approximate functional equation As is well known, it is possible to approximate the Riemann zeta function, using two Dirichlet polynomials rather than one, in a way which allows shorter polynomials, namely: 522 for x y : , and s in any fixed vertical strip away from the pole at Here appears in the functional equation It is likely that there should be an analogue to our Theorem for approximate functional equations of this type stating that such an approximate functional equation can only hold for L(s) in the range T t 2T provided that xy > At first we hoped that our method, based on counting zeros, would lead to this result, but were stopped by the following example which shows that the analogue for Proposition 1 (at least in its obvious form) does not hold. Take L(s) to be ~(s); take x = y = 1. Then the \"approximation\" is 1 + x(s) and, despite the fact that x and y are bounded, this has asymptotically (in fact, with an error term only O(log T)) as many zeros as ~(s) itself inside the rectangle 0 Q 1, T t 2T. EXAMPLE 3. Existence of approximations It is well-known that smoothed truncations of a Dirichlet series can provide very good approximations. Let u(x) be a C°° function with compact support in (0, 1], such that -

Abstract: © Annales de l’institut Fourier, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: Let k be an algebraically closed field and f e k[x1 . x,+]. Fix an embedded resolution h: X -An+I of f-I {O} and denote by Ei, i E S, the irreducible components of h-l(f-l{O}) with multiplicity Ni in the 0 0 visor of f o h. Put also Es := Ei \ Uj1iEj, and denote by X(Ei) its Euler characteristic. Several conjectures concerning Igusa's local zeta function and the topological zeta function of f motivate the study of Euler characteristics associated to subsets UiE TEi of UiEsE1 , which are maximal connected with respect to the property that dINi for all i E T. Here d e N, d > 1 . We prove that if h maps UiET Ei to a point, then

Abstract: The generalized Igusa local zeta function $Z_{\Q _p}(s)$ associated to $(\SL_7,\rho)$, where $\rho$ is the Cartan product of the first, third and fifth fundamental representations of $\SL_7$, is explicitly computed and shown not to satisfy the expected functional equation $$Z_{\Q _p}(s)|_{p \mapsto p^{-1}}=p^{-7s}Z_{\Q _p}(s).$$

Abstract: The objective of this announcement is the statement of some recent results on the classification of generalized Igusa local zeta functions associated to irreducible matrix groups. The definition of a simple Igusa local zeta function will motivate a complete classification of certain generalized Igusa local zeta functions associated to simply connected simple Chevalley groups. In addition to the novelty of these results are the various methods used in their proof. These methods include use of the concept of canonical basis from quantum group theory and a formula expressing Serre’s canonical measure μc in terms of a suitably normalized Haar measure μ and density function Φ. The relevance of these results in the general theory of Igusa local zeta functions is also discussed.

Abstract: We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray and Singer and the holomorphic $L^2$-torsion, where the latter is defined via the $L^2$-theory of Atiyah. For higher fundamental rank twisted torsion numbers appear.

Abstract: In this paper we prove the trace formulas for the Reidemeister numbers of group endomorphisms in the following cases:the group is finitely generated and an endomorphism is eventually commutative; the group is finite ; the group is a direct sum of a finite group and a finitely generated free abelian group; the group is finitely generated, nilpotent and torsion free . These results had previously been known only for the finitely generated free abelian this http URL a consequence, we obtain under the same conditions on the fundamental group of a compact polyhedron,the trace formulas for the Reidemeister numbers of a continuous map and under suitable conditions the trace formulas for the Nielsen numbers of a continuous map.The trace formula for the Reidemeister numbers implies the rationality of the Reidemeister zeta function. We prove the rationality and functional equation for the Reidemeister zeta function of an endomorphisms of finitely generated torsion free nilpotent group and of a direct sum of a finite group and a finitely generated free abelian group.We give a new proof of the rationality of the Reidemeister zeta function in the case when the group is finitely generated and the endomorphism is eventually commutative and in the case when group is finite.We give also another proof for the positivity of the radius of convergence of the Nielsen zeta function and propose an exast algebraic lower bound estimation for the radius.We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free abelian group with the Lefschetz zeta function of the unitary dual map,and as consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion.

Abstract: Summary. In this paper, we discuss some questions about the Riemann zeta function (ζ), with historical remarks, particularly about the link between the zeta function and the prime numbers, based on a famous Eulerian p roduct (1737). In 1 859, Riemann defined the zeta function for complex numbers; he c onjectured that all nontrivial zeroes of ζ: x→ζ(x) are on the line Re(x) = Ω; this hypothesis has never been proved; computations of the zeroes of ζ have been made, and the first 1,500,000,001 nontrivial zeroes of ζ are on the line Re(x) = Ω (1986).